Question

Based on information from the Bureau of Labor Statistics, of women in the civilian labor force, 51.8% are married, 28.2% have never been married, 12.8% are divorced, 4.1% are separated, and 3.1% are widowed. Of the married women, 3.6% are unemployed; of the never-married women, 8.5% are unemployed; of the divorced women, 5.2% are unemployed; of the separated women, 8.3% are unemployed; of the widowed women, 5.7% are unemployed.(a) What is the probability a randomly selected woman from the civilian labor force is unemployed?
(b) If an unemployed woman is selected, what is the probability she is married?
(c) If an unemployed woman is selected, what is the probability she never married?
(d) If an unemployed woman is selected, what is the probability she is divorced?
(e) If an unemployed woman is selected, what is the probability she is separated?
(f) If an unemployed woman is selected, what is the probability she is widowed?

Answer 1

Answer:

(a) 0.054

(b) 0.3413

(c) 0.4439

(d) 0.1233

(e) 0.0630

(f) 0.0327

Step-by-step explanation:

Let's denote the events as follows:

M = a woman in the civilian labor force is married

N = a woman in the civilian labor force is never married

D = a woman in the civilian labor force is divorced'

S = a woman in the civilian labor force is separated

W = a woman in the civilian labor force is widowed

U = a woman in the civilian labor force is unemployed.

The provided information is:

P (M) = 0.518           P (U|M) = 0.036

P (N) = 0.282           P (U|N) = 0.085

P (D) = 0.128            P (U|D) = 0.052

P (S) = 0.041            P (U|S) = 0.083

P (W) = 0.031           P (U|W) = 0.057

(a)

The law of total probability states that:

$P (A)=P(A|B)P(B)+P(A|C)P(C)$

Compute the probability a randomly selected woman from the civilian labor force is unemployed as follows:$P(U)=P(U|M)P(M)+P(U|N)P(N)+P(U|D)P(D)+P(U|S)P(S)+P(U|W)P(W)\\=(0.036\times0.518)+(0.085\times0.282)+(0.052\times0.128)+(0.083\times0.041)+(0.057\times0.031)\\=0.054444\\\approx0.054$Thus, the probability of a woman being unemployed is 0.054.

(b)

Compute the probability that an unemployed woman is married:

$P (M|U)=\frac{P(U|M)P(M)}{P(U)} =\frac{0.036\times0.518}{0.054}=0.3413$

Thus, the probability that an unemployed woman is selected is married is 0.3413.

(c)

Compute the probability that an unemployed woman is never married:

$P (N|U)=\frac{P(U|N)P(N)}{P(U)} =\frac{0.085\times0.282}{0.054}=0.4439$

Thus, the probability that an unemployed woman is selected is never married is 0.4439.

(d)

Compute the probability that an unemployed woman is divorced:

$P (D|U)=\frac{P(U|D)P(D)}{P(U)} =\frac{0.052\times0.128}{0.054}=0.1233$

Thus, the probability that an unemployed woman is selected is divorced is 0.1233.

(e)

Compute the probability that an unemployed woman is separated:

$P (S|U)=\frac{P(U|S)P(S)}{P(U)} =\frac{0.083\times0.041}{0.054}=0.063$

Thus, the probability that an unemployed woman is selected is separated is 0.0630.

(f)

Compute the probability that an unemployed woman is widowed:

$P (W|U)=\frac{P(U|W)P(W)}{P(U)} =\frac{0.057\times0.031}{0.054}=0.0327$

Thus, the probability that an unemployed woman is selected is widowed is 0.0327.